Abstract
Recently we formulated the Coulomb problem in a rotationally invariant NC configuration space specified by NC coordinates <em>x<sub>i</sub>, i</em> = 1, 2, 3, satisfying commutation relations<em> [x<sub>i</sub>, x<sub>j</sub> ] = 2iλε<sub>ijk</sub>x<sub>k</sub></em> (<em>λ</em> being our NC parameter). We found that the problem is exactly solvable: first we gave an exact simple formula for the energies of the negative bound states <em>E<sup>λ</sup><sub>n</sub></em> &lt; 0 (n being the principal quantum number), and later we found the full solution of the NC Coulomb problem. In this paper we present an exact calculation of the NC Coulomb scattering matrix <em>S<sup>λ</sup><sub>j</sub> (E)</em> in the <em>j</em>-th partial wave. As the calculations are exact, we can recognize remarkable non-perturbative aspects of the model: 1) energy cut-off — the scattering is restricted to the energy interval 0 &lt; <em>E</em> &lt; <em>E</em><sub>crit</sub> = 2/<em>λ</em><sup>2</sup>; 2) the presence of two sets of poles of the S-matrix in the complex energy plane — as expected, the poles at negative energy <em>E</em><sup>I</sup><sub><em>λ</em>n</sub> = <em>E</em><sup><em>λ</em></sup><sub>n</sub> for the Coulomb attractive potential, and the poles at ultra-high energies <em>E</em><sup>II</sup><sub><em>λ</em>n</sub> = <em>E</em><sub>crit</sub> − <em>E<sup>λ</sup></em><sub>n</sub> for the Coulomb <em>repulsive</em> potential. The poles at ultra-high energies disappear in the commutative limit <em>λ</em>→0.
Highlights
The basic ideas of non-commutative geometry have been developed in [1] and in a form of matrix geometry in [2]
In [9] we gave an exact simple formula for the NC negative bound state energies, and in [10] we presented the full solution of the NC Coulomb problem
As the model is exactly solvable, we were able to find an exact formula for the NC Coulomb scattering matrix Sjλ(E) in j-th partial wave
Summary
The basic ideas of non-commutative geometry have been developed in [1] and in a form of matrix geometry in [2]. The analysis performed in [3] led to the conclusion that quantum vacuum fluctuations and Einstein gravity could create (micro)black holes which prevent localization of space-time points. This requires non-commutative (NC) coordinates xμ in space-time satisfying specific commutation relations. Later in [4] it was shown that such field theories in NC spaces can emerge as effective low energy limits of string theories. These results supported a vivid development of non-commutative QFT.
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